MULTIPLICATION OF SCHWARTZ DISTRIBUTIONS AND COLOMBEAU GENERALIZED FUNCTIONS

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1 Journl of Applied Anlysis Vol. 5, No. 2 (1999), pp MULTIPLICATION OF SCHWATZ DISTIBUTIONS AND COLOMBEAU GENEALIZED FUNCTIONS B.P. DAMYANOV eceived Jnury 6, 1999 nd, in revised form, April 23, 1999 Abstrct. The differentil C-lgebr G( m ) of generlized functions of J.- F. Colombeu contins the spce D ( m ) of Schwrtz distributions s C-vector subspce nd the notion of ssocition in tht lgebr generlizes the equlity in D ( m ). This is prticulrly useful for evlution of distribution products, s they re embedded in G( m ), in terms of distributions gin. The pper is devoted to results on prticulr products of distributions with coinciding singulrities, s well s to generl property of Colombeu product of distributions together with some pplictions. All formuls obtined, when restricted to dimension one, re esily trnsformed into the setting of regulrized model products in clssicl distribution theory. 1. Introduction The fmous problem of multipliction of Schwrtz distributions hs been for long time n objective of mny reserch studies. This is due to the lrge 1991 Mthemtics Subject Clssifiction. Primry 46F10. Key words nd phrses. Multipliction of Schwrtz distributions, Colombeu generlized functions. Prtilly supported by NSF Grnt φ 610 of the Bulgrin Ministry of Science nd Eduction. ISSN c Heldermnn Verlg.

2 250 B.P. Dmynov employment of distributions in the nturl sciences nd other mthemticl fields where products of distributions with coinciding singulrities often pper. Strting with the historiclly first construction of distributionl multipliction proposed by König [8] nd the sequentil pproch developed by Mikusiński nd co-uthors [2], there hve been numerous ttempts to define products for the distributions, or rther to enlrge the rnge of existing products (see [12] for complete review nd bibliogrphy). Moreover, severl ttempts hve been mde to include the distributions into lgebrs of generlized functions with the differentition lwys possible nd subject to the Leibniz rule, or else into differentil lgebrs. According to the clssicl Schwrtz counter-exmples, however, in ssocitive lgebrs of generlized functions, multipliction nd differentition cn not simultneously extend the corresponding clssicl opertions unrestrinedly. One thus needs to limit the requirements on the multipliction. Most complete list of such properties so fr possesses the ssocitive differentil lgebr of generlized functions of J.- F. Colombeu [3]. The distributions re linerly embedded in tht lgebr nd the multipliction is comptible with the opertions of differentition nd products with C -differentible functions. Unlike the nonstndrd-nlysis pproch, the generlized-number system of Colombeu, tht includes infinitely smll nd lrge quntities, is not bsed on non-constructive methods (nd the xiom of choice). Yet, when it comes to ppliction to the nturl sciences, one needs results in terms of clssicl quntities. The ssocition process in Colombeu lgebr, providing fithful generliztion of the equlity of distributions in D ( m ), enbles us to obtin results in terms of distributions (nd numericl fctors) the so-clled Colombeu products. With the pplictions in mind, we follow this pproch here: we evlute prticulr products of distributions with coinciding singulrities, s embedded in Colombeu lgebr, in terms of ssocited distributions. By this, the results obtined cn be reformulted s regulrized model products in the clssicl distribution theory, pplicble to the mny-vrible cse s well. We recll now the following well known result in dimension one obtined by Mikusiński in [11] : x 1. x 1 π 2 δ 2 (x) = x 2, x. (1) Although neither of the products on the left-hnd side of (1) exists, their sum still hs correct mening in D (). Another formul of tht type in dimension one in nonstndrd pproch to distribution theory ws given in [13] : H. δ (x) + δ 2 (x) = δ (x) /2. (2)

3 Multipliction of Schwrtz distributions nd Colombeu functions 251 H denotes here the Heviside function, nd = stnds for the equlity up to n infinitesiml quntity; which cn be thought of s nonstndrd nlogue of Colombeu ssocition. Similrly, we will derive severl formuls of distributionl products in dimension one, where the individul summnds do not hve ssocited distribution, but their sum considered s single entity dmits such distribution. We think it relevnt to nme them products of Mikusiński type. 2. Definitions nd preliminry results We now recll the fundmentls of Colombeu theory, following its presenttion in [12, Ch.3]. Nottion. IfN 0 stnds for the nonnegtive integers nd p = (p 1, p 2,..., p m ) is multiindex in N m 0, we let p = m i=1 p i nd p! = p 1!...p m!. Then, if x = (x 1,..., x m ) is in m, denote x p = (x p 1 1, xp 2 2,..., xpm m ) nd p = p / x p xpm m. Also, by x < 0 is ment: x 1 0,..., x m 0 nd x 0. Further, if q is inn 0, we put A q () = {ϕ(x) D() : xj ϕ(x)x = δ0j for 0 j q, where δ 00 = 1, δ 0j = 0 for j > 0}. This lso extends to m s n m-fold product: A q ( m ) = {ϕ(x) D( m ) : ϕ(x 1,..., x m ) = m i=1 χ(x i) for some χ in A q () }. We finlly denote ϕ = m ϕ( 1 x) for ϕ in A q ( m ) nd > 0. Definition 1. Let E [ m ] be the set of functions f(ϕ, x) : A 0 ( m ) m C tht re C -differentible with respect to x by fixed prmeter ϕ; which is C-lgebr with the pointwise function opertions. Ech generlized function of Colombeu is then n element of the quotient lgebr G( m ) = E M [ m ] / I [ m ]. Here the sublgebr E M [ m ] of E [ m ] is the set of moderte functions f(ϕ, x) in E [ m ] such tht for ech compct subset K of m nd ny p in N m 0 there is q in N such tht : for ech ϕ in A q ( m ) there re c > 0, η > 0 stisfying sup x K p f(ϕ, x) c q for 0 < < η. In turn, the idel I [ m ] of E M [ m ] is the set of functions f(ϕ, x) such tht for ech compct subset K of m nd ny p inn m 0 there is q in N such tht : for every r q nd ech ϕ in A r ( m ) there re c > 0, η > 0 stisfying sup x K p f(ϕ, x) c r q, for 0 < < η. The lgebr G( m ) contins the distributions on m, cnoniclly embedded s C-vector subspce by the mp i : D ( m ) G( m ) : u ũ = [ũ(ϕ, x)]. The representtives here re ũ(ϕ, x) = (u ˇϕ)(x) where ˇϕ(x) = ϕ( x) nd ϕ is running the set A q ( m ). We note tht similr, yet different schemes of new generlized functions were introduced by Antonevich nd dyno [1] nd by Egorov [5].

4 252 B.P. Dmynov Definition 2. A generlized function f in G( m ) is sid to dmit some u in D ( m ) s n ssocited distribution, which is denoted by f u, if f hs representtive f(ϕ, x) in E M [ m ] such tht for ech ψ(x) in D( m ) there exists q in N 0 such tht, for ll ϕ(x) in A q ( m ), lim 0 m f(ϕ, x)ψ(x)dx = u, ψ. (3) This definition is independent of the representtive chosen nd the distribution ssocited, if it exists, is unique; the imge in G( m ) of every distribution is ssocited with the ltter [12]. The -ssocition is thus generliztion of the equlity of distributions. Then by Colombeu product we men the product of some distributions s they re embedded in Colombeu lgebr G( m ) whenever the result dmits n ssocited distribution (see [10] for comprison with other distribution products). The following coherence result holds [12, Proposition 10.3] : If the regulrized model product of two distributions exists, then their Colombeu product lso exists nd coincides with the former. We now recll two lemms proved in [4] tht will be needed to prove our min results. The following seprtion property is pplicble to distributions hving tensor product structure. Lemm 1. Let u nd v be distributions in D ( m ) such tht u(x) = m i=1 ui (x i ), v(x) = m i=1 vi (x i ) with ech u i nd v i in D (), nd suppose tht their embeddings in G() stisfy ũ i.ṽ i w i, for i = 1,..., m. Then ũ.ṽ w, where w = m i=1 wi (x i ) is in D ( m ). We will lso employ the following. Lemm 2. For n rbitrry ϕ in A 0 (), i.e. ϕ in D() with ϕ(t) dt = 1, suppose tht supp ϕ [, b], for some, b in. Then, for n rbitrry p in N 0, it holds ϕ(t) t (y t) p ϕ (p) (y) dy dt = ( 1)p p! 2. (4) 3. Min results on Colombeu products of distributions The -ssocition is consistent with the C ( m ) liner opertions nd the differentition: f u implies p f p u nd.f.u for n rbitrry p in N m 0 nd in C ( m ) [12, 3.10]. However, only wek vrint of the Leibniz rule (or conditionl Leibniz rule ) cn be pplied to the derivtives i = / x i (i = 1,..., m) of Colombeu product of distributions, s shown by the next.

5 Multipliction of Schwrtz distributions nd Colombeu functions 253 Proposition 1. (i) Let the embeddings of the distributions u nd v into G( m ) stisfy ũ. ṽ w, where w is in D ( m ). Then it holds i u. ṽ + ũ. i v i w, i = 1, 2,..., m. (5) (ii) If moreover ũ. i v dmits some w 1 in D ( m ) s ssocited distribution, then the Colombeu product i u. ṽ exists for ech i = 1, 2,..., m nd it holds i u. ṽ i w w 1. (6) emrk 1. In generl, only the sum on the left-hnd side of (5) hs n ssocited distribution, but not the individul summnds in it. Proof. (i) Since for ech ϕ(x) in A 0 ( m ) the corresponding representtives re smooth functions of x by fixed, we get by integrting by prts for n rbitrry test-function ψ(x) in D( m ) nd i = 1,..., m: i ũ(ϕ, x)ṽ(ϕ, x)ψ(x)dx = ũ(ϕ, x) i (ṽ(ϕ, x)ψ(x)) dx m m = ũ(ϕ, x) i ṽ(ϕ, x)ψ(x)dx ũ(ϕ, x)ṽ(ϕ, x) i ψ(x)dx. m m Further we tke into ccount tht in the differentil lgebr E M [ m ] of representtives it holds, by definition ( see [3, Ch. 3] ) : i f(ϕ, x) = i [ x f(ϕ, x) ]; which pplied to the embedding of distributions gives : i ũ = i u for ll u in D ( m ). Thus we cn write i u(ϕ, x)ṽ(ϕ, x)ψ(x)dx + ũ(ϕ, x) i v(ϕ, x)ψ(x)dx m m = ũ(ϕ, x)ṽ(ϕ, x) i ψ(x)dx. (7) m Now, by (3) nd on multiple differentition in D (), eqution (7) gives ( ) lim i u(ϕ 0, x)ṽ(ϕ, x)ψ(x)dx + ũ(ϕ, x) i v(ϕ, x)ψ(x)dx m m = lim 0 m ũ(ϕ, x)ṽ(ϕ, x) i ψ(x)dx = w, i ψ = i w, ψ. This proves eqution (5). (ii) In this prticulr cse, eqution (7) yields lim 0 m i u(ϕ, x)ṽ(ϕ, x)ψ(x)dx + lim 0 m ũ(ϕ, x) i v(ϕ, x)ψ(x)dx = i w, ψ.

6 254 B.P. Dmynov Hence, by ssumption, lim x u(ϕ 0, x)ṽ(ϕ, x)ψ(x)dx = i w, ψ w 1, ψ. m This proves the existence of ssocited distribution for i u. ṽ, s well s eqution (6). The proof is complete. We now proceed to prticulr Colombeu products of distributions. First we recll result proved in [4] which will be needed in the sequel. Proposition 2. For n rbitrry p in N m 0, let p δ(x), x p + nd x p be the embeddings in G( m ) of the distributions p δ(x), x p + = {xp for x 0, = 0 elsewhere } nd x p = { ( x)p for x 0, = 0 elsewhere }. Then it holds x p +. p δ(x) ( 1) p p! 2 m δ(x), xp. p δ(x) p! 2 m δ(x). (8) emrks 2. Equtions (8) re known in distribution theory, lthough being only derived s regulrized products by the prticulr choice of symmetric mollifiers (the regulrizing functions). The eqution x p δ (p+q) (x) = ( 1)p (p + q)! q! δ (q) (x) (p, q N 0 ), (9) is esily shown to hold in D (). Now equtions (8), when restricted to one vrible, re consistent N with (9) by q = 0, in view of the identity x p = x p + + ( 1)p x p (p 0). Products of the form x p q δ(x) (x in m ) re useful in quntum renormliztion theory in Physics, nd we next prove prticulr result on such products. Proposition 3. For n rbitrry p in Nm, let x p nd ^ p 1 δ(x) be the embeddings in G( m ) of the distributions x p nd p 1 δ(x). Then it holds x p.^ p 1 δ(x) ( 1) p (p 1)! 2 m (2p 1)! 2p 1 δ(x). (10) Proof. Consider first the one-vrible cse : x, p N. By definition, we hve x p = ( 1) p 1 /(p 1)! d p /dx p (ln x ). We therefore represent the first multiplier in (10) by x p (ϕ, x) = ( 1)2p 1 (p 1)! p+1 ( ) y x ln y ϕ (p) dy.

7 Multipliction of Schwrtz distributions nd Colombeu functions 255 Differentiting in D (), we get for the representtive of ^δ(p 1) (x) ^δ (p 1) (ϕ, x) = ( 1) p 1 p δ y, ϕ (p 1) ((y x)/) = ( 1) p 1 p ϕ (p 1) ( x/). Now if supp ϕ(x) [, b] for some, b in, then supp ϕ( x/) [ b, ]. Thus, for ech ψ(x) in D(), replcing (y x)/ = v nd x/ = u, we get V p := x p (ϕ, x)^δ(p 1) (ϕ, x), ψ(x) = = ( 1) p (p 1)! 2p+1 ( 1) p (p 1)! 2p 1 b ψ(x)ϕ (p 1) ( x ψ( u) ϕ (p 1) (u)du ) dx b+x +x ( ) y x ln y ϕ (p) dy ln ( v u ) ϕ (p) (v)dv. By Tylor s theorem (with some η in (0, 1) ) nd chnging the order of integrtion which is permissible here we obtin V p = 2p 1 k=0 + O() =: 2p 1 k=0 ( 1) p+k ψ (k) (0) (p 1)! k! 2p k 1 ϕ (p) (v)dv ϕ (p) (v)dv ( 1) p+k ψ (k) (0) (p 1)! k! 2p k 1 I k + o (1). ln( u v )ϕ (p 1) (u)u k du (ln + ln u v ) ϕ (p 1) (u) u 2p ψ (2p) (ηu)du Here we hve tken into ccount tht lim 0+ p. ln = 0 for ech p > 0. This gives for the Lndu symbols: O(). ln = o (1). Note tht the ltter term is multiplied by n expression which includes definite integrls mjorizble by constnt. Further integrtion by prts the integrted term being zero ech time then gives for ech k = 0, 1,..., 2p 1 : (k + 1) I k = = = ϕ (p) (v)dv dv ϕ (p) (v)dv dv ln ( u v )ϕ (p 1) (u)d(u k+1 v k+1 ) ϕ (p) (v)(u k+1 v k+1 ) ln ( u v )ϕ (p) (u)du ϕ (p 1) (u) (uk+1 v k+1 ) du u v k f(v, u)du J p,l J p 1,k l, l=0

8 256 B.P. Dmynov where J p,l := ϕ(p) (v) v l dv. The first term bove is zero since its integrnd, denoted by f(v, u), stisfies f(u, v) = f(v, u). Thus we hve V p = 2p 1 k=0 k l=0 ( 1) k+p 1 ψ (k) (0) (p 1)!k!(k + 1) 2p k 1 J p,l J p 1,k l + o (1). A direct clcultion of the integrls J m,n yields: J m,n = 0 if m > n, nd J m,m = ( 1) m m!. However, if l > p in the first integrl, then k l < p 1 in the second one. Thus the only non-zero term in the sum bove is obtined when l = p nd k l = p 1, or else, by k = 2p 1. Hence V p = ( 1)p (2p 1)! J p,pj p 1,p 1 ψ (2p 1) (0) + o (1) = ( 1)p (p 1)! δ (2p 1) (x), ψ(x) + o (1). 2(2p 1)! Therefore pssing to the limit, s 0, we obtin eqution (10) for m = 1 nd n rbitrry p in N. In the mny-vrible cse, due to the tensor-product structure of the distributions in considertion, we cn pply Lemm 1 tht yields m x p.^ p 1 δ(x) = x m ( p i i.^ δ(p i ( 1) p i ) 1) (p i 1)! (x i ) δ (2pi 1) (x i ) 2(2p i 1)! i=1 which completes the proof. i=1 = ( 1) p (p 1)! 2 m 2p 1 δ(x); (2p 1)! emrks 3. Eqution (10) is derived in [6] nd [9] under (different) dditionl requirements on the mollifiers. We note lso tht when restricted to one vrible, the Colombeu products given in this section re esily trnsformed into regulrized model products of distributions. Indeed, replcing ϕ(x) ρ ( x) with ϕ in A 0 () which requirement on ϕ we hve only used nd employing (3) we get for ll ψ in D() : lim 0 ũ(ϕ,x)ṽ(ϕ,x), ψ(x) = lim 0 (u ρ )(v ρ ), ψ = [u,v], ψ, where ρ stisfies exctly the requirements imposed on the mollifiers for generl model products of distributions [12, 2.7]. 4. Applictions to the one dimensionl cse We now proceed to some results on Mikusiński type products of distributions in the one-dimensionl cse. So, if we pply the wek Leibniz rule (5) to Colombeu product given by eqution (10) in the one-vrible cse, we thus obtin the following.

9 Multipliction of Schwrtz distributions nd Colombeu functions 257 Corollry 1. For n rbitrry p in N, the embeddings in G() distributions x p nd δ (p) (x) stisfy x p. δ (p) (x) p ^x p 1.^δ(p 1) (x) ( 1)p (p 1)! 2(2p 1)! In the prticulr cse p = 1, it holds δ (2p) (x). of the (11) x 1. δ (x) x 2. δ(x) 1 2 δ (x). (12) Denoting now Ȟ := H( x)( = x0 ), one esily checks tht (x ) = Ȟ nd (Ȟ) = δ. Then the result below follows on pplying the wek Leibniz rule to equtions (8) considered in dimension one. Corollry 2. For n rbitrry p in N 0, it holds in G() : x p +. ^δ(p+1) (x) + p x p 1 +. δ (p) (x) ( 1) p p! 2 δ (x). (13) x p. ^δ(p+1) p 1 (x) p x. δ (p) (x) p! 2 δ (x). (14) In the cse p = 0 of (8), eqution (5) yields : H. δ (x) + δ 2 (x) 1 2 δ (x), Ȟ. δ (x) δ 2 (x) 1 2 δ (x). (15) emrk 4. The first eqution bove coincides with (2), but we hve used no uxiliry requirements on the mollifiers such s to be even functions, s required in [13]. We cn derive more formuls of Mikusiński type. Indeed, equtions (13) nd (14) by p = 1 red : x +. δ (x) + H. δ (x) 1 2 δ (x), x. δ (x) Ȟ. δ (x) 1 2 δ (x). Combining these equtions with the ones in (15), we get x +. δ (x) δ 2 (x) δ (x), x. δ (x) δ 2 (x) δ (x). (16) Furthermore, s shown by the next two propositions, equtions (15) nd (16) concerning the distributions x p ± nd δ(p+1) (x) cn be directly generlized for ny p in N0.

10 258 B.P. Dmynov Proposition 4. For n rbitrry p in N0, the embeddings in G() of the distributions x p + nd δ(p+1) (x) stisfy ( 1) p x p +.^δ(p+1) (x) + p! δ 2 (x) (p + 1)! 2 δ (x). (17) Proof. First of ll, for ech ψ(x) in D(), we get on the chnge x/ = t : δ 2 (ϕ, x), ψ(x) = 1 ( 2 ϕ 2 x ) ψ(x) dx = 1 ϕ 2 (t) ψ( t) dt p+2 = ψ(0) b ϕ 2 (t) dt ψ (0) t ϕ 2 (t) dt + O(). (18) Denoting V p := x p + (ϕ, x).^δ(p+1) (ϕ, x), ψ(x), we get further ( ) b V p = ( 1)p+1 (x + t) p ϕ(t)dt ϕ (p+1) ( x ) ψ(x) dx b x/ = 1 = ψ(0) + ψ (0) ψ( y) ϕ (p+1) (y)dy ϕ(t)dt ϕ(t)dt t t y (y t) p ϕ(t)dt (y t) p ϕ (p+1) (y)dy y (y t) p ϕ (p+1) (y)dy + O() =: ψ(0) J 1 + ψ (0) J 2 + O(). On multiple integrtion by prts, the integrted term being zero ech time, we get J 1 = ( 1) p p! ϕ dt t ϕ (y) dy = ( 1) p p! Also, Lemm 2 nd multiple integrtion by prts give J 2 = Hence ϕ(t)dt t (y t) p+1 ϕ (p+1) (y)dy + = ( 1) p+1 (p + 1)! ( 1)p p! ( ψ(0) ( 1) p V p = p! 1 2 (p + 1)! ψ (0) + O(). t ϕ 2 (t)dt. ϕ 2 (t) dt ψ (0) tϕ(t)dt t ϕ 2 (t) dt. (y t) p ϕ (p+1) (y)dy ) t ϕ 2 (t) dt

11 Multipliction of Schwrtz distributions nd Colombeu functions 259 Tking now into ccount eqution (18), we get ( 1) p x p + (ϕ, x).^δ(p+1) (ϕ, x), ψ(x) + p! δ 2 (ϕ, x), ψ(x) (p + 1)! = δ (x), ψ(x) + O(). 2 Therefore pssing to the limit, s 0, we obtin eqution (17) for ech p in N. The proof is complete. Proposition 5. For n rbitrry p in N0, the embeddings in G() of the distributions x p nd δ(p+1) (x) stisfy x p.^δ(p+1) (x) p! δ 2 (x) (p + 1)! 2 δ (x). (19) Proof. Since x p = ( x)p + for ny p in N 0, eqution (19) is obtined on replcing x x in (17) nd tking into ccount tht δ (p+1) ( x) = ( 1) p+1 δ (p+1) (x). emrk 5. Equtions (17) nd (19) in G() re esily shown to be consistent with eqution (9) by q = 1, which holds in the spce D (). Consider finlly the even nd odd sums of the distributions x p +, xp, x in nd p inn0, s defined in [7, 1.3]) : x p = x p + +xp, x p sgn x = x p + xp. Combining now equtions (17) nd (19) we obtin the following. Corollry 3. Let x p nd x ^ p sgn x be the embeddings in G() of the distributions x p nd x p sgn x (p in N0). Then it holds : x p.^δ(p+1) (x) (p + 1)! δ (x), p = 0, 2, 4,... x p.^δ(p+1) (x) 2 p! δ 2 (x) 0, p = 1, 3, 5,... x ^ p sgn x.^δ(p+1) (x) + 2 p! δ 2 (x) 0, p = 0, 2, 4,... x ^ p sgn x.^δ(p+1) (x) (p + 1)! δ (x), p = 1, 3, 5,.... (20) (21) (22) (23)

12 260 B.P. Dmynov emrk 6. Observe tht equtions (20) nd (23) give ordinry Colombeu products of distributions, while the other two bove s it is with ll the equtions (11) (19) re of Mikusiński type. eferences [1] Antonevich, A., Y dyno, On generl method of constructing lgebrs of generlized functions, Soviet. Mth. Dokl. 43(3) (1991), [2] Antosik P., Mikusiński, J., Sikorski,., Theory of Distributions, Elsevier Sci. Publishing, Amsterdm, [3] Colombeu, J.-F., New Generlized Functions nd Multipliction of Distributions, North Hollnd Mth. Stud. 84, Amsterdm, [4] Dmynov, B., esults on Colombeu product of distributions, Comment. Mth. Univ. Crolin. 43(4) (1997), [5] Egorov, Y., On the theory of generlized functions, ussin Mth. Surveys 43(5) (1990), [6] Fisher, B., The product of distributions, Qurt. J. Mth. Oxford Ser. (2) 43 (1971), [7] Gel fnd, I., Shilov, G., Generlized Functions, Vol. 1, Acdemic Press, New York, [8] König, H., Neue Begründung der Theorie der Distribution, Mth. Nchr. 9 (1953), [9] Itno, M., emrks on the multiplictive products of distributions, Hiroshim Mth. J. 6 (1976), [10] Jelínek, J., Chrcteriztion of the Colombeu product of distributions, Comment. Mth. Univ. Crolin. 43(2) (1986), [11] Mikusiński, J., On the squre of the Dirc delt-distribution, Bull. Polish Acd. Sci. Mth. 43 (1966), [12] Oberguggenberger, M., Multipliction of Distributions nd Applictions to Prtil Differentil Equtions, Longmn, Essex, [13] ju, C., Products nd compositions with the Dirc delt function, J. Phys. A 43(2) (1982), Blgovest P. Dmynov Bulgrin Acdemy of Sciences INNE Theory Group Tsrigrdsko Shosse Sofi, Bulgri dmynov@bg400.bg